Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type

Abstract

We determine the exact asymptotic order of the entropy numbers of compact embeddings B s 1This complements the known results for weights of polynomial type. The estimates are given in terms of the number 1/p = 1/p 1 − 1/p 2 and the function w(x). We find an interesting new effect: if the growth rate at infinity of w(x) is below a certain critical bound, then the entropy numbers depend only on w(x) and no longer on the parameters of the two Besov spaces. All results remain valid for Triebel-Lizorkin spaces as well. Mathematics Subject Classification (2000) 41A46 · 46E35

!Entropy Numbers of Embeddings of Weighted Besov Spaces. Ii

“…What is the exact asymptotic entropy behaviour of the embeddings of the corresponding weighted Besov spaces? In the forthcoming paper [24] this question is solved for logarithmic weights, and in this situation some new phenomena occur.…”

Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights

“…V, §3, Thm. 9]), and were continued and extended by Kühn, Leopold, Sickel and the second author in the series of papers [21,22,23,33]. As an application one obtains spectral estimates of certain pseudo-differential operators in the way of the program proposed by Edmunds and Triebel [10].…”

Wavelet Frames, Sobolev Embeddings and Negative Spectrum of Schrödinger Operators on Manifolds with Bounded Geometry

“…The same theorem in [19] asserts the embedding q 1 (2 jδ p 1 ( w)) → q 2 ( p 2 ) is compact if and only if (4.8) holds and in addition…”

“…The assumption (4.23) implies (4.12) so the estimate from above can be proved in the same way as for α (v −1 (t)). Now we used Proposition 4 in [19] (with d = 1 as above).…”

“…Moreover, we give estimates from below and above of the asymptotic behavior of entropy numbers of the above compact embeddings. We use the estimates for sequence spaces developed in [17][18][19]. Roughly speaking if the quotient w 1 /w 2 is sufficiently large at infinity then the entropy numbers e k behave like k −(s 1 −s 2 )/d on the other hand if w 1 /w 2 is sufficiently small at infinity then the asymptotic behavior of the entropy numbers depends on the quotient.…”